[Paper Review] A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

Jacob Bedrossian, Alex Blumenthal+1 more|arXiv (Cornell University)|Jul 30, 2020

Stochastic processes and financial applications102 references22 citations

TL;DR

This paper introduces a novel regularity-based method to establish quantitative lower bounds on the top Lyapunov exponent for weakly-dissipative, weakly-forced stochastic differential equations (SDEs). By deriving a Fisher information identity linking the Lyapunov exponent to the stationary density of the projective process and applying a new quantitative hypoelliptic regularity theory in L1, the authors prove the first mathematically rigorous positivity of the top Lyapunov exponent for the Lorenz-96 model in any dimension when driven by additive noise on any consecutive pair of modes.

ABSTRACT

In this article, we review our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations. This hallmark of chaos has long been observed in these models, however, no mathematical proof had been made for either deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an $L^1$-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies H\"ormander's condition. We review the recent contributions of the first and third author on the verification of this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio. Finally, we briefly contrast this work with our earlier work on Lagrangian chaos in the stochastic Navier-Stokes equations. We end the review with a discussion of some open problems.

Motivation & Objective

• To develop a mathematically rigorous method for establishing quantitative lower bounds on the top Lyapunov exponent in stochastic systems where existing methods fail.
• To prove the positivity of the top Lyapunov exponent for weakly-dissipative, weakly-driven SDEs, including fundamental models like Lorenz-96 and Galerkin-truncated Navier-Stokes equations.
• To establish a new connection between the Lyapunov exponent and the regularity of the stationary density of the projective process on the unit tangent bundle.
• To develop a quantitative hypoelliptic regularity theory in an L1 framework that controls degenerate Fisher information from below using Sobolev norms.

Proposed method

• Derive a new identity (Theorem A) expressing the top Lyapunov exponent λ1 in terms of a degenerate Fisher information functional FI(f) of the stationary density f on the unit tangent bundle SRn.
• Establish a quantitative hypoelliptic regularity result (Theorem B) showing that the Fisher information FI(f) controls the local W^{s,1}_{loc} Sobolev norm of f for some s > 0.
• Use the projective process (wt) = (xt, vt) on SRn, where vt tracks the normalized tangent direction of the flow, to analyze the dynamics of linearized perturbations.
• Lift the original vector fields Xk on Rn to projective vector fields ˜Xk on SRn using the Sasaki metric and orthogonal decomposition into horizontal and vertical components.
• Apply a novel L1-based hypoelliptic regularity theory to bound the Fisher information from below by a local Sobolev seminorm, thereby linking regularity of the stationary density to lower bounds on λ1.
• Prove that the Lie algebra generated by the projective vector fields ˜Xk satisfies the parabolic Hörmander condition, ensuring hypoellipticity and ergodicity of the projective process.

Experimental results

Research questions

• RQ1 Can a new method be developed to rigorously establish lower bounds on the top Lyapunov exponent for weakly-dissipative, weakly-forced SDEs?
• RQ2 Does the Fisher information of the stationary density on the projective bundle control the Lyapunov exponent, and can this be quantified?
• RQ3 Can a quantitative L1 hypoelliptic regularity theory be constructed to bound the Fisher information from below using Sobolev norms?
• RQ4 Is the top Lyapunov exponent positive for the Lorenz-96 model in any dimension under additive noise on any consecutive pair of modes?
• RQ5 Can this method be applied to other fundamental models such as Galerkin-truncated Navier-Stokes equations?

Key findings

• The paper establishes a new identity (Theorem A) showing that the top Lyapunov exponent satisfies nλ1 − 2λΣ = FI(f), where FI(f) is a degenerate Fisher information functional on the stationary density f of the projective process on SRn.
• The authors prove that the Fisher information FI(f) is bounded from below by a local W^{s,1}_{loc} Sobolev seminorm of f for some s > 0, establishing a quantitative link between regularity and Lyapunov exponent (Theorem B).
• For the Lorenz-96 model in any dimension n ≥ 3, the top Lyapunov exponent is strictly positive when the additive noise is applied to any consecutive pair of modes, marking the first rigorous proof of chaos in this model.
• The method applies to a broad class of SDEs with bilinear drift that preserve volume and norm, including Galerkin-truncated Navier-Stokes equations, for which positivity of the Lyapunov exponent is also rigorously established.
• The analysis confirms that the projective process on SRn is ergodic and satisfies the parabolic Hörmander condition under mild assumptions, ensuring the existence and uniqueness of the stationary measure.
• The Furstenberg-Khasminskii formula is rigorously re-derived in the projective setting, confirming that the sum Lyapunov exponent λΣ equals the integral of the divergence of the drift field, and that nλ1 − 2λΣ equals the integral of the projective divergence of the noise fields.